ABSTRACT
This chapter is concerned with the constructive theory as it results from modern mathematics. The Infinitesimal Calculus is the traditional name for the differential and integral calculus together. The so-called infinitesimal calculus, therefore, has nothing to do with the infinitesimal, and has only indirectly to do with the infinite—its connection with the infinite being, that it involves limits, and only infinite series have limits. The philosophical theory of the Calculus has been, ever since the subject was invented, in a somewhat disgraceful condition. Leibniz himself—who, one would have supposed, should have been competent to give a correct account of his own invention—had ideas, upon the topic, which can only be described as extremely crude. The differential coefficient depends essentially upon the notion of a continuous function of a continuous variable.
